Journal of Nanomaterials

Volume 2015 (2015), Article ID 697596, 7 pages

http://dx.doi.org/10.1155/2015/697596

## Effect of Nanostructure on Thermal Conductivity of Nanofluids

Department of Chemical Engineering, Pennsylvania State University, University Park, PA 16802, USA

Received 26 May 2015; Revised 5 August 2015; Accepted 24 August 2015

Academic Editor: Wei Chen

Copyright © 2015 Saba Lotfizadeh and Themis Matsoukas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The presence of colloidal particles is known to increase the thermal conductivity of base fluids. The shape and structure of the solid particles are important in determining the magnitude of enhancement. Spherical particles—the only shape for which analytic theories exist—produce the smallest enhancement. Nonspherical shapes, including clusters formed by colloidal aggregation, provide substantially higher enhancements. We conduct a numerical study of the thermal conductivity of nonspherical structures dispersed in a liquid at fixed volume fraction in order to identify structural features that promote the conduction of heat. We find that elongated structures provide high enhancements, especially if they are long enough to create a solid network (colloidal gel). Cross-linking further enhances thermal transport by directing heat in multiple directions. The most efficient structure is the one formed by hollow spheres consisting of a solid shell and a core filled by the fluid. In both dispersed and aggregated forms, hollow spheres provide enhancements that approach the theoretical limit set by Maxwell’s theory.

#### 1. Introduction

Common fluids used in heating/cooling processes have very low thermal conductivity in comparison to solid materials [1]. Adding nanoparticles results in considerable improvement of the fluid thermal properties [2, 3]. While solid particles of any size will increase the thermal conductivity of the base fluid, there is a great interest in nanoparticles due to practical considerations associated with the production of stable suspensions that resist precipitation. Adding to this practical concern is a number of experimental reports of unusually large but often inconsistent increases of the thermal conductivity when working with nanoparticles [3], which have motivated various hypotheses as to the microscopic origins of these behaviors [4, 5]. Large enhancements of the thermal conductivity have recently been associated with aggregation [6–8]; however, analytical theories are not equipped to address the conductivity of aggregated structures quantitatively. The standard theoretical tool for conduction in inhomogeneous media such as colloidal dispersions is Maxwell’s mean field theory [9]. Maxwell’s result, originally developed in the context of electrical conduction, gives the conductivity of a dispersion of spheres in a continuous medium: where is the conductivity of the dispersion, is the conductivity of the particles, is the conductivity of the fluid, and is the volume fraction of the particles. As the conductivity of solids is typically higher to much higher than that of liquids, this equation predicts that the conductivity of the dispersion is higher than that of the fluid. An even higher conductivity is obtained if the roles of the solid and liquid are inverted to produce a dispersion of liquid droplets inside a continuous solid matrix with the same volume fraction. The result is obtained by swapping and in (1) and replacing by : Equations (1) and (2) establish two limits for the thermal conductivity of an inhomogeneous system composed of two phases at fixed volume fraction. The lower limit, (1), refers to a dispersion of the more conductive phase in a medium of low conductivity, whereas (2) refers to a dispersion of the less conductive phase in a continuum formed by the more conductive phase. Eapen et al. [7] suggested that (2) can be viewed as an upper limit for a colloidal gel, whose structure consists of a continuous solid network with regions of liquid dispersed in the interior. The two bounds may be taken then to represent the limits of fully dispersed particles (lower limit) and a fully gelled colloid (upper limit), with finite clusters falling in the space between these bounds. Recently, Lotfizadeh et al. [8] confirmed this hypothesis by showing that the thermal conductivity of a suspension at fixed volume fraction of primary particles increases monotonically with cluster size and reaches the upper limit of Maxwell’s theory in the gel state. In a subsequent study, this behavior was quantified via an analytic model based on Maxwell’s theory [10]. This model, however, depends on a parameter that reflects the structure of the cluster and which cannot be obtained by analytic means.

Structural details of colloidal clusters are important in determining the conductivity of the dispersion. While Maxwell’s theory provides a baseline calculation for two idealized limits, that of fully dispersed and that of fully gelled states, for nearly all other cases theory is inadequate and one must resort to numerical simulations. To evaluate thermal conductivity of colloidal suspensions with different properties including particles shape and size along with investigating the effect of aggregation and cluster structure using model configurations, we need to go beyond experimental limitations and employ an accurate numerical model to avoid colloidal complications. In macroscopic simulations, large-scale structural effects can be captured. These simulations can be done by different standard models to solve the macroscopic conduction equation; however Monte Carlo method is both fast and accurate and especially well suited for complex geometries.

In this study we present a systematic investigation on the thermal conductivity of nonspherical particles with special interest in identifying structures that maximize conductivity at fixed volume fraction of the solid and approach the upper limit of the theory. We explore the range of validity of Maxwell’s theory for different particle shapes, evaluate thermal conductivity of solid particles, hollow particles as well as rods, and other nonspherical shapes, and identify the structures that produce maximum enhancement at fixed volume fraction of the dispersed phase.

#### 2. Monte Carlo Method for the Thermal Conductivity of Clusters

Of the several methods available for the conductivity of heterogeneous structures, Monte Carlo is particularly useful because it allows the study of systems with arbitrarily complex geometries. In Monte Carlo we obtain the thermal conductivity of a two-phase system via the statistics of a biased random walk along sites with different thermal conductivities. The method as implemented here ignores the motion of the particle through the fluid medium as well as all fluid-mediated interactions between particles. We justify omitting these factors for two reasons. The first one is that Maxwell’s theory itself only considers conduction and neglects all other mechanisms of heat transfer. In this respect the simulation provides a direct comparison to the predictions of Maxwell’s theory. The second reason is based on the previous experimental and theoretical studies that show the enhancement of the thermal conductivity of clustered dispersions is fully captured by the conduction of heat along the solid backbone of the cluster and that other mechanisms, if present, make contributions that are at best within the error bounds of the experimental measurements [6–8, 10–13].

The Monte Carlo method used here is based on the work of Van Siclen [14]. The volume of the system (cluster in a fluid) is discretized in cubic elements of equal size, each element representing either fluid or particle. Heat walkers are launched randomly inside the system and take unit steps in one of 6 directions that exist in the discretized 3 lattice chosen at random. Movement of the walker is biased by the conductivity of the sites the walker is leaving from and moving to with transition probability: where is the conductivity of the current site and the conductivity of the neighbor. Time is advanced by and the process is repeated to produce a trajectory in time, . The thermal diffusivity of the composite material is obtained from the Einstein relationship [15]: and the conductivity is finally calculated from its relationship with the thermal diffusivity, , where is the heat capacity. The implementation of the method is straightforward even for very complex solid structures and has been used by many others in the study of the thermal conduction in inhomogeneous media [15–19].

The dimensionless parameter that controls the conductivity of the suspension is the ratio of the solid-to-liquid thermal conductivity. Oxide materials in typical fluids have conductivity ratios in the range 2–50, and metallic particles can reach values of the order of 100. The ratio is trivial and in this case the conductivity of the suspension is identical to that of the fluid for all particle structures and solid volume fractions. In the limit , heat conduction is entirely governed by the least conducting phase (liquid) and is independent of the conductivity of the solid [10, 20]. In this study we use ratios in the range 2 to 50, roughly corresponding to a range between aqueous dispersions of silica (lower limit) and alumina (upper limit). The results are qualitatively very similar to other ratios.

#### 3. Results and Discussion

##### 3.1. Validation of Numerical Simulation against Maxwell’s Theory

An important element of the simulation is the discretization size of a unit element relative to the size of the spherical particle. In general, the smaller the discretization size, the more accurate the simulation but also more computationally intensive. It is common to represent primary solid particles by a single lattice site [11, 21]; however, the numerical accuracy of this simplification has not been reported in the literature. Thus the first test is evaluating the thermal conductivity of a single sphere within a unit lattice as a function of the discretization size. The results are shown in Figure 1. As the number of sites increases, the thermal conductivity of the dispersion, reported as a ratio over the conductivity of the fluid (we refer to this ratio as enhancement), increases and converges to the value predicted by Maxwell’s theory. The most coarse representation of the sphere by a single lattice site underestimates the conductivity of the dispersion by 7%. Approximating the sphere with 7 sites (three orthogonal rows of 3 sites, each with a common center) produces results with the same accuracy as a sphere composed of 100 sites. In all subsequent simulations we use at least 7 lattice sites to represent each primary particle. As further validation we compute the conductivity of the dispersion as a function of volume fraction up to a maximum fraction of 25%, shown in Figure 2. These results are in excellent agreement with Maxwell’s theory and further corroborate the findings of Belova and Murch [21] who reported very good agreement between simulation and Maxwell’s theory for all volume fractions up to the point that particles begin to touch.